Well, lets make sure we understand what the term "expand" means. When things expand the particles they are composed of move apart from each other. A good example of this is baking a loaf of bread in an oven. There is escaping carbon dioxide gas which causes the loaf of bread to expand, or rise. Traces of this gas may be seen in the infrastructure of the bread, with its porous texture. So to expand does not necessarily involve adding more matter. In the case of the expanding universe, it is thought the universe was created by a "big bang", which hurled the pieces of existing matter throughout the universe. Galaxies, which are huge collections of stars, are moving away from each other, similar to the way pieces of an explosion would move away from each other. So it is possible to have an expanding universe without having to add more matter to it.
It is absulutely possible, the expansion of the universe occurs. The matter in our cosmos move apart from each other.
We can say that our earth is moving away from the nearby celestial bodies because the universe is expanding.
When we blow a balloon, it expands. So, why it expands, it is the same reason that the matter stretches.
So, I think that one day the cosmos might stop expanding.